Subdominant Galois Resultants, Partition Moment and Cumulant Generating Functions, and Rank-One Hankel Time-Reversal Identities for Generalized k-Step Fibonacci Recurrences

Let {F_n^(k)} (for n in Z and k >= 2) denote the standard k-step Fibonacci (k-bonacci) sequence with initial conditions F₋₍ₖ₋₂₎^(k) = ... = F_0^(k) = 0 and F_1^(k) = 1, and let α_k in (1, 2) be the unique Pisot–Vijayaraghavan root of the characteristic polynomial P_k(x) = x^k - x^(k-1) - ... - x - 1. Decomposing F_n^(k) = A_n + D_n into its dominant Binet mode A_n = α_k^(n+k-2) / P_k'(α_k) and its subdominant Galois residual D_n = Σᵣ₌₁ᵏ⁻¹ β_r^(n+k-2) / P_k'(β_r), we establish five main results: 1. Universal Perturbation Annihilation and Quadratic Convergence: For the (k+1)-term exponential equation b^(Fₙ₊ₖ^(k) x_n) - Σₘ₌₀ᵏ⁻² c_m* b^(Fₙ₊ₘ^(k) x_n) - μ* = 0 (with Σₘ₌₀ᵏ⁻² c_m* = 1 and b > 1), we prove that all k - 1 first-order subdominant Galois error modes vanish simultaneously if and only if the fixed-point partition weights satisfy q_m*(k) = (α_k^(k-1-m) - 1) / (α_k^k - 1) and q_μ*(k) = (k - 1) / (α_k^k - 1). This yields universal quadratic convergence x_n / xₙ₊₁ - α_k = α_k (Φ_n^(2) - Φₙ₊₁^(2)) + O(δ_n^3). For k = 3 (Tribonacci), we prove the quadratic partition syzygy [q_μ*(3)]^2 = 2 q_0*(3) q_1*(3) in Z[α_3] and derive both the instantaneous binary quadratic form in (δ_n, δₙ₊₁) and the phase-averaged secular convergence constant C̄_3 = w*(3) / (α_3^2 [P_3'(α_3)]^2). 2. Partition Power Sums, Moment Generating Functions, Cumulants, and Asymptotics: We derive exact closed-form reductions in Q(α_k) for all partition power sums S_p(k) = Σₘ₌₀ᵏ⁻² q_m*(k) α_k^(pm) (p >= 0), the q-Lambert resummed Moment Generating Function G_k(t), all raw moments M_r(k) = Σₘ₌₀ᵏ⁻² q_m*(k) (α_k^k - α_k^m)^r (r >= 1), and the shift-invariant cumulants κ_r(k) of K_k(t) = ln G_k(t) (including κ_2(3) = 2 / (α_3 + 2)^2, γ_1(3) = -α_3(α_3 - 1) / √2, κ_2(k) ~ 2^k / 6, and γ_1(k) ~ -(3√6 / 7) 2^(k/2)). These exact moments yield sharpened three-term asymptotic expansions for w*(k) ~ (k - 1) / 4^k, μ*(k) ~ (k - 1) / 2^k, and C̄_k^can ~ (k - 1) / (4 · 16^k) as k -> ∞. 3. Subdominant Galois Resultant Identity: We prove that the (k-1) × (k-1) Hankel determinant of the subdominant residual D_n satisfies det H_n^(k)(D) = (-1)^(C(k,2) + (k-1)n) / (α_k^(n+k-2) P_k'(α_k)). 4. Rank-One Hankel Time-Reversal Identities: Via the rank-one perturbation H(F) = H(D) + A_n v_α v_α^T and the Matrix Determinant Lemma, we prove the integer Hankel time-reversal identity det H_n^(k)(F) = (-1)^(C(k,2) + (k-1)n) F₋ₙ₋₂ₖ₊₄^(k) and the k × k topological invariant det Hn,k(F) = (-1)^(C(k,2) + (k-1)n). 5. Backwards k-bonacci Generating Functions and Diophantine Zero Classifications: We show that the time-reversed sequence F₋ₙ₋₂ₖ₊₄^(k) has rational generating function z(1 - z) / (1 - 2z^k + z^(k+1)), grows asymptotically at rate ρ_k ~ 3^(1 / (k + 1/3)), and has complete integer zero sets n in {-2, -1, 2, 15} for k = 3, n in {-4, -3, -2, 1, 2, 6} for k = 4, n in {-6, -5, -4, -3, 0, 1, 2, 6, 7, 12} for k = 5, and C(k, 2) = k(k - 1)/2 zeros for every even k >= 2. ---MSC 2020 Classification: Primary 11B39, 11R06, 11C20; Secondary 11D61, 11B83, 60E10, 37C25.arXiv Subject Classes: math.NT (Number Theory); math.DS (Dynamical Systems); math.RA (Rings and Algebras).DOI: https://doi.org/10.6084/m9.figshare.34800135